


W ' . "? 



A HANDBOOK 

ON THE 



TEETH OF GEARS, 

THEIR CURVES, PROPERTIES 

AND 

PRACTICAL CONSTRUCTION. 




WITH 

ODONTOGRAPHS, 

FOR BOTH EPICYCLOIDAL AND INVOLUTE TEETH, 

RULES FOR THE STRENGTH OF TEETH, 

A TABLE OF PITCH DIAMETERS, 

AND MUCH OTHER 

GENERAL INFORMATION 

ON THE SUBJECT. 



GEORGE B. GRANT, 

ee BEVERLY STREET, BOSTON, MASS. 



PRICE, ONE DOLLAR. 



Copj-right, 1885, by Geo. B, Grant. 



LIBRARY OF CONGRESS. 

mpf — iijpri5¥ 1" -- 

SheK_.:..(i.'I.C 
tsi? 



CNITED STATES OF AMERICA. 



A HANDBOOK 

ON THE 



TEETH OF GEARS, 

THEIR CURVES, PROPERTIES 

AND 

PRACTICAL CONSTRUCTION. 





WITH 



ODONTOGRAPHS, 

FOR BOTH EPICYCLOIDAL AND INVOLUTE TEETH, 

RULES FOR THE STRENGTH OF TEETH, 

A TABLE OF PITCH DIAMETERS, 

AND MUCH OTHER 

GENERAL I N F O R M A T I _.. _ ^ 

ON THE SUBJECT. /^v? ^ '^f.aP^RHIMr. ^^T, 

GEORGE b/grant, \ MAR 11 188] 

63 BEVERLY STREET, BOSTON, ^>^&By ^ ^^^ ^ ^^y^^^ 

PRICE, ONE DOLLAR. 
Copyright, 1885, by Geo. B. Grant. 



tK 



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&l. 



1^ 



OV-357% 



THE TEETH OF 

GEAR WHEELS 



INTRODUCTION. 

Few meclianical subjects have attracted the attention of scientific men 
to such an extent, or are so intimately connected with mathematics, as tlie 
proper construction of tlie teetli of gear wheels, and, as a consequence, few 
can show such an advance as has here been made, from the rough cog 
wheel of not many years ago, to the neat cut gear of the present day. 

It is not apparent wherein much further improvement is needed in our 
knowledge of the theory of the subject, but it is evident that much remains 
to be done towards its practical application, and to induce the working 
mechanic to understand and use the improvements that have been developed 
by the mathematician and the inventor. The theory seems to be full and 
well nigh perfect, but the mill-wright and the machinist still clings to 
imperfect rules and clumsy devices that should have been forgotten years 
ago, and few workmen have a clear Jcnowledge of even the rudiments of the 
science which it is their business to apply to practical purposes. 

It is the mathematical and scientific character of the subject that makes 
it so difi&cult to the j^ractical man, who can understand but little of it as it 
is commonly presented in elaborate treatises or encyclopaedias, and who 
takes but little interest in the study of a matter that bristles with strange 
characters and technical terms. 

I have here undertaken to address the workman as well as the man of 
science, and have felt obliged to leave out nearly everything that cannot be 
treated in a i)lain, descriptive manner, to use language that any intelligent 
man can understand, and to refer to more pretentious works than this for 
demonstrations, or unessential details. 

A volume of a thousand pages would not properly present the whole 
subject, and this little pamphlet can deal only with the main principles and 
prominent points. It is not a treatise, it is a hand-book that does not 
pretend to cover the whole ground, and its principal object is to present 
the new odontographs, which I believe to be superior to those heretofore 
in use for the i)urpose of designing the teeth of gear wheels. 

FIRST PRINCIPLES. 

The original gear wheel had pins or projections for 
teeth, of any form that would serve the general purpose 
ar.d communicate an unsteady motion from one wheel to 
another. 

HG. 1. 
THE ORIGINAL GEAR WHEEL. 

The perfect gear wheel is the friction wheel, communicating 
a smooth, uniform, rolling motion, by means of the f rictional 
j contact of its surface. It is, in fact, a gear wheel with a 
great many very small, weak, and irregular teeth. 
The whole aim and object of the science of the teeth of gear 
wheels is to increase the size and strength of these teeth with- 
out destroying the uniformity of the motion they transmit, 
and this is accomplished by studying the shape of the teeth, and giving their 
bearing surfaces the curved outline that is found to produce the desired 
result. 

There are an infinite number of curves that will meet the requiremeht, 
but only two, the epicycloid and the involute, are of any practical impor- 
tance, or in actual use. 





FRICJION VVHEE-S. 



THE EPICYCLOIDAL TOOTH. 

The epicycloidal or double curve tooth 
has its bearing surface formed of two 
curves, meeting at the pitch line P, 
which corresponds to the working cir- 
cle of the perfect gear wheel of fig. 2. 

If a small circle5a,be rolled around on 
the outside of the pitch circle, p, a fixed 
^ tracing point, a, in its edge, will trace 
• out the dotted line called an epicycloid, 
•and a small part of this curve near the 
I pitch line, usually one sixth of its full 
iiy/ height,forms the face of the tooth. 

Similarly, if a small circle, B, be rolled 
around on the inside of the pitch line, 
its tracing point, b, will describe the 
internal epicycloid, or hypocycloid, a 
small portion of which is used for the 
flank of the tooth. 







FIG. 3. THE EPICYCLOIDAL TOOTH. 




FIG .4 WHOLE EPICYCLOIDAL TEETH. 



If a projection be formed on the friction 
wheel fig. 4, the curved outline of which 
is a whole epicycloid E, and a depression 
be formed in the wheel N having a whole 
hypocycloid H for its outline, then, if both 
curves have been formed by the same 
describing circle B, it can be mathemati- 
cally demonstrated that the two curves 
will just touch and slide on each other, 
without separating or intersecting, while 
the two friction wheels roll together. 

The reverse of this fact is also true; that, 
if one wheel drives another by means of 
an epicycloidal projection on it working 
against a hypocycloidal depression in the 
other, both curves being formed by the 
same describing circle, tlie two wheels will 
roll together as uniformly as if driven by 
frictional contact, and it is this peculiar property of the epicycloid that gives 
it its value for the purpose in hand. 

The pressure acting between the two curves is in the direction of the 
line dg, is direct only at the start, and becomes more and more oblique, 
until, when the middle points, q q, come together, and beyond, there is no 
driving action at all. This defect forbids the use of the whole curve and we 
can use but a small portion of it near the pitch line. Another projection and 
depression must be formed so near the first that they will come into work- 
ing position before the first pair are out of contact, thus forming the theo- 
retically perfect but incomplete gears of fig. 5. 

Practical requirements still further 
modify the apparent shape of the 
tooth, for it is desirable that the 
wheels shall work in either direction, 
and that they shall be interchangea- 
ble, so that any one of a set of several 
shall work with any other of that set. 
This can be accomplished only *by 
making the curves face both ways. 
and by putting both projections and 
depressions on each gear, thus form- 
ing the familiar tooth of fig. 3. 




FIQ. 5. INCOMPLETE EPICYCLOIDAL TEETH. 



THE INTERCHANGEABLE SET. 

If all the curves of a set of several gears, both the faces and the flanks 
of each gear, are described by the same rolling circle, the set will be 
interchangeable, and any one vrill work perfectly with any other. 

This is a property of the greatest practical importance, and interchangea- 
ble sets should come into as universal use on heavy mill work as with cut p-ear- 
ing. It is the only system that will allow the use of a set of ready made 
cutters, and is therefore essential to the economical manufacture of cut 
gear wheels. 

The diameter of the rolling circle is usually made half the diameter of the 
smallest gear of the set, and that gear will have straight radial lines for 
flanks. 

The set in almost universal use and adopted for all the odontographs, has 
twelve teeth in its smallest gear, but there is a tendency to change this well 
established system, and create confusion for which the writer can see no 
adequate excuse, by the adoption of a pinion of fifteen teeth as the base 
or smallest gear. It may be admitted that as large a base as possible should 
be used, but the change from twelve to fifteen seems to be unwarranted 
in view of the confusion it creates by the abrupt change from an old and 
good rule to a new one that is a mere shade better, and the trouble it makes 
with small pinions of eight to twelve teeth. 

RADIAL FLANK TEETH. 

If the internal curves, or flanks, of a pair of gears that are to run together 
are on each radial straight lines described by a rolling circle of half its 
pitch diameter, and the rolling circle that describes the flanks of one gear 
is used to describe the faces of the other gear, then, the two gears will form 
a pair fitted to each other and not interchangeable with other gears. 

This style of gear is very often used under the erroneous impression that 
it is the best possible form, and will give the least possible friction and 
thrust on the bearings, but the saving in friction over the interchangeable 
form would be an exceedingly difficult thing to measure by any practicable 
method, although it can be mathematically demonstrated to be a fact, and 
the slender roots of such teeth make them weaker and much inferior to the 
others. The odontograph figures show both a pair of these gears, and the 
same pair on the interchangeable plan, also, by the dotted lines on the former 
figure, the shapes as they would be on the interchangeable plan. It is plainly 
seen that the interchangeable faces are but a shade more rounding, while 
their flanks are so curved that the teeth are much stronger at the roots. The 
larger the describing circle, the less the theoretical thrust and friction, and 
if the flanks Avere formed by a describing circle of more than half the diam- 
eter of the gear, the teeth would be undercurved, the friction less, and their 
strength less, than that of the radial flank tooth. 

In practical matters it is a good plan to give flrst place to practical points, 
and not to take too much notice of minute theoretical advantages, and 
there is no good reason, that will bear the test of experiment, for adopting 
the radial flank, non-interchangeable, and weak tooth, in preference to the 
strong tooth of the interchangeable system. 

THE PITCH. 

The pitch is a term used to designate the size of the tooth, and is either 
circular or diametral. 

THE CIRCULAR PITCH or more properly the circumferential pitch, 
is the actual distance from tooth to tooth measured along the curve of the 
pitch line, and is expressed in inches, as f inch pitch, 1| inch pitch, etc. 

The table gives the proper pitch diameter of a gear of any given number 
of teeth, and one inch circular pitch. The tabular numbers must be multi- 
plied by any other pitch that is in use. 

Formerly, the circular pitch was the only one known, but it has deser- 
vedly gone out of use on cut gears, and it is hoped may soon be abandoned 
altogether. It is a clumsy, awkward, and troublesome device on either large 
or small work, having its origin in the ignorance of the past, and owing its 



existence not to any perceptible merit, but to habit, and the natural per- 
sistence of an established custom. 

With the circular pitch the relation between the pitch diameter of the 
gear, and the number of teeth on it, is fractional. If the diameter is a 
convenient quantity, such as a whole number of inches, the pitch must be 
an inconvenient fraction, and if the pitch is a handy part of an inch, the 
diameter will contain an unhandy decimal. 

With the circular pitch there is no one length of tooth that is better than 
any other, and consequently there is no agreement upon that point. Each 
maker is at liberty to chose his own distance at random, and whatever he 
chose s is as good as any other. 

Its worst feature is that it leads to endless errors, for the average mechanic 
appreciates convenience more than accuracy, and will stretch his figures to 
suit his facts, with a botch as the common result. 

A millwright figures out a diameter of 22.29 inches for a gear of one inch 
pitch and 70 teeth, and failing to make such a clumsy figure fit his work or 
his foot rule, and thinking a quarter of an inch or so to be of no importance, 
he lets it go at 22 whole inches. The same process on its mate of 15 teeth 
gives a 5 inch gear instead of one of 4.78 inches diameter, and the pair will 
never run or wear together properly. His only alternatives are to adopt 
the clumsy true diameters, or else use the clumsy figure .988 inch for his 
pitch. 

Again, he is apt to apply a carpenter's rule directly to the teeth of the 
gear he is to repair or match, and naturally takes the nearest convenient 
fraction of an inch as his measurement, when the real pitch may be just 
enough different to spoil the job. 

There is no reason whatever for using the circular pitch, unless the work 
to be done is to match work already in use. 

THE DIAMETRAL PITCH is an immense improvement on the old 
fashioned circular pitch. It is not a measurement, but a number, or ratio. 
It is the number of teeth on the gear, for each inch of its pitch diameter, 
and its merit is that it establishes a convenient and manageable relation 
between these two principal elements, so that the calculations are of the 
simplest description and the results convenient and accurate. 

The product of the pitch and the pitch diameter is equal to the number of 
teeth, and the number of teeth divided by the pitch is equal to the pitch diame- 
ter. A gear of 15 inches diameter and 2 pitch has 30 teeth, and a gear of 27 
teeth of 4 pitch has a pitch diameter of 6f inches. 

The rule that the length of the tooth is two pitch parts of an inch, | or i 
an inch for 4 pitch, f or 1 inch for 2 pitch, etc. is so simple and so much bet- 
ter than any other that it is never disputed, and is in universal use. 

The circular and diametral pitches are connected by the relation 

cXp=3.1416. 

or, the product of the circular and the diametral pitch is the number 3.1416. 

THE ADDENDUM. 

For reasons expressed above we can use but a small part of the epicy- 
cloidal curve near the pitch line, limiting it by a circle drawn at a distance 
inside or outside of the pitch line called the addendum. The outside limit 
need not be the same as the inside limit, but it is customary to make them 
equal. 

When the diametral pitch is used, the length of the addendum is always 
one pitch part of an inch, as Jthinch for 4 pitch, ^rd inch for 3 pitch, etc. If 
we use the same proportion for circular pitches the addendum will be -^ xir^ 
circular pitch, and the value ird of the circular pitch may be adopted as the 
most convenient for use. 

THE CLEARANCE. 

Theoretically, the depression formed inside the pitch line should be only as 
deep as the projection outside of it is high, but to allow for practical defects 
in the making or in the adjustment of the teeth, and to provide a place for 



dirt to lodge, the depression is always deeper than theory requires by an 
amount called the clearance. The amount of the clearance is arbitrary, but 
the sixteenth part of the depth of the tooth is a convenient and customary 
measure, or -i^th of the circular pitch, and 1 divided by 8 times the diametral 
pitch. The following tables will be convenient and save calculation : 

CLEARANCE FOR CIRCULAR PITCHES. 



Circular pitch. 
Clearance. 


.02 .03 


f 
.03 


.04 


1 

.04 


.05 


.05 


If H 

.06 .06 


If 
.07 


2 
.08 


2i 
.09 


.10 


8 
.12 


CLEARANCE FOR DIAMETRAL PITCHES. 


Diametral pitch. 
Clearance. 


6 

.02 


5 
.03 


4 
.03 


H 

.04 


.04 


3 
.04 


2f^ 
.05 


2i 
.05 


.06 


2 
.06 


If 
.08 


.09 


H 1 

.10 .12 



THE BACKLASH. 

When wooden cogs or rough cast teeth are used, the inevitable irregular- 
ities require that the teeth should not pretend to fit closely, but that the 
spaces should be larger than the teeth by an amount called the backlash. 
The amount of the backlash is arbitrary, but it is customary to make it 
about equal to the clearance. 

Cut gears should have no allowance for backlash, and involute teeth need 
less backlash than epicycloidal teeth. 



PITCH DIAMETERS. 

TT-OR ONE HSrCH CIRCTJLA.R FITCH. 

For Any Other Pitch, Multiply by that Pitch. 



T. 


P.D. 


T. 


P.D. 


T. 


P.D. 


T. 


P.D. 


]0 


3.18 


33 


10.50 


66 


17.83 


79 


25.15 


11 


3.50 


34 


10.82 


57 


18.15 


80 


25.47 


12 


3 82 


35 


11.14 


68 


18.47 


81 


25.79 


13 


4.14 


36 


11.46 


69 


18.78 


82 


26.10 


14 


4 46 


37 


11.78 


60 


19.10 


83 


26.43 


15 


4.78 


88 


12.10 


61 


19.42 


84 


26.74 


16 


6.09 


39 


12.42 


62 


19.74 


85 


27.06 


17 


6.40 


40 


12.74 


63 


20.06 


86 


27.38 


18 


6.73 


41 


13.05 


64 


20.38 


87 


27.70 


19 


6.05 


42 


13.37 


65 


20.69 


88 


28.02 


20 


6.37 


43 


13.69 


66 


21.02 


89 


28.:^4 


21 


6 69 


44 


14.00 


67 


21.33 


90 


28.65 


22 


7.00 


45 


14.33 


68 


21.65 


91 


28.97 


23 


7.32 


46 


11.65 


69 


21.97 


92 


29.29 


24 


7.64 


47 


14.96 


70 


22.29 


93 


29.60 


25 


7.96 


48 


15.28 


71 


22^.60 


94 


29.93 


26 


8.28 


49 


15.60 


72 


22.92 


9.i 


30.25 


27 


8 60 


50 


15.92 


73 


23.24 


96 


30.56 


28 


8.90 


51 


16.24 


74 


23.56 


97 


30.88 


2d 


923 


52 


16 56 


75 


23.88 


98 


31.20 


30 


9.55 


53 


16.87 


76 


24.20 


99 


31.52 


31 


9.87 


54 


17.19 


77 


24.52 


100 


31.84 


32 


10.19 


65 


17.62 


78 


24.83 







6 




riG. 6. THE EPICYCLOID. 




THE EPICYCLOID. 

THEORETICAL FORMATION. 

The true epicycloid, shown by fig. 6, 
is perpendicular to the pitch line at 
the origin a, and forms an endless 
series of lobes about it, as in fig, 3. 

The most convenient and simple 
process for draAving it, is to step it off 
with the dividers. Several describing 
circles, M^ to M'^, are drawn at ran- 
dom ; steps are made, as shown by 
the figure, from the origin a^ to past 
each tangent point, a^ to a^, and then 
the same number back, around each 
circle, to locate the several points, b^ 
to b^, on the curve, which is then 
drawn by hand through the points, 
and is accurately in place if the steps 
are sinall. 
By the mechanical method for drawing the 
curve, the describing circle, B, is rolled around 
the pitch circle A, and a tracing point or pencil 
P, draws the curve. A steel ribbon s, is fastened 
to the templets at each end, and assists in keep- 
ing them in place. 

This process is the main principle of the epicy- 
cloidal engine, which carries a scribing tool, or 
a rotary cutter at p, to trace or cut out a tem- 
plet that is then used in forming gear teeth or 
gear cutters. 

It is, of course, the most accurate method 
known, but it is not available for ordinary pur- 
poses, for unless the templets are well made and 
skillfully handled, the resulting curve will be 
poorly drawn, and the method, although simple in principle, may be consid- 
ered difficult in its practical application. 

F^^ACTICAL FORMATION. 

Of course nothing but the perfect curve will answer its purpose with per- 
fect accuracy, but the epicycloid is a peculiar curve which cannot be accu- 
rately drawn by any simple process, or with common instruments, particu- 
larly when the teeth are small, and it is customary to use arcs of circles or 
other curves, which approximate as nearly as possible to the true curve. 

Such an arc can be made to agree with the curve so closely that it is a need- 
less refinement to be more particular for most practical purposes, such as 
drafting teeth, making wooden cogs or patterns for cast teeth, or even the 
templets for shaping gear cutters and planing bevel gear teeth. 

Some makers of rough cast or heavy planed gearing go to great expense 
to construct the (supposed to be) theoretically true epicvcloid, by means of 
rolling circles. This practice looks very much indeed like accuracy, but if 
he had an absolutely true curve as a templet, supposing he could make such 
a thing, the maker of this class of work could not produce from it a work- 
ing tooth more nearly perfect than if the templet was properly constructed 
of circular arcs. It is labor lost to lay out teeth to the thousandth of an 
inch, that must be constructed with ordinary hand or machine tools, or 
shaped with a chisel and mallet. 

Furthermore, it is a question if the delicate processes and epicycloidal 
engines used for the finest cut gear work, can serve practical purposes and 
construct templets to work from, better than intelligent and skillful 
hand-work. It is a fact that the best work in this line is made from tem- 
plets that are laid out by theory, but dressed into shape and perfected I y 
hand and eye processes. 



FIG. 7. THE EPICYCLOIDAL ENGINE. 



ODONTOGRAPHS. 

^Nlaiiy arbitrary or " rule of thumb " methods for shapinor p^ear teeth have 
been proiX)secl, but they are p:enerally worthless, and reliance should be 
r>1aced only on such as are founded on the mathematical prniciples of the 
curve to be imitated. Of these only three are known to the writer. 




THE WILLIS ODONTOGRAPH is a method for finding the 
center m of the circle which is tangent to the epicycloid a b c, at the point b, 
where it is cut by a line bm, which passes through the adjacent pitch jjoint 
k, and makes the angle gkf=T5° with the radial line kf. 

The radius used, is not the line m b, 1-ut the more convenient line m a. 

The instrument is nothing Avhatever but a piece of card or sheet metal 
cut to the angle ef 75°, which is laid against the radial line kf, as a guide 
for drawing the line k m. The center distance k m, to be laid off along the 
line thus drawn is given by a table that accompanies the instrument. 

Xo instrument is necessary, for the line k m may be placed by drawing the 
arc f g with a radius of one inch, and laying off the chord fg=l. 22 inch. The 
tabular distance km can be readily computed from 



, c 

Ki nil — 2Q g 


t 

t+12 


k^ m^ = 2^^3 


t 

t-12 



in which c is the circular pitch in inches, and t is the number of teeth in 
the gear. , 

The Willis odontograph, as found in use, is confined to the single case of 
an interchangeable series rmming from twelve teeth to a rack, but for any 
possible pair of gears the angle becomes 

g k f = 90° — i§i° 



and 



ki nil = 
k2 m2 = 



s c 
.628 
sc 



t-f s 

t 



sill. 



sill. 



180° 

s 
180° 



.628 t — s s 

in which t is the number of teeth in the gear being drawn and s the number 
in the mate. 
The accuracy of the Willis circular arc will be examined further on. 



8 



THE IMPROVED WILLIS ODONTOGRAPH. 

EPICYCLOIDAL TEETH. 
TWELVE TO RACK. INTERCHANGEABLE SERIES. 







FOR ONE 




For one inch 


NUMBER OF 


DIAMETRAL PITCH. 


CIRCULAR PITCH. 


TEETH 


For any otliei 


L" pitch, 


divide 


For any other pitch, mul- 


IN THE GE^E. 


by that pitch. 




tiply by that pitch. 






Faces. 


Flanks. 


Faces. 


Flanks. 1 


Exact. 


Intervals. 


Rad. 


Dis. 


Rad. 


Dis. 


Rad. 

.73 


Dis. 


Rad. 


Dis. 


12 


12 


2.30 


.15 


oo 


oo 


.05 


OO 


OO 


m 


13-14 


2.35 


.16 


15.42 


10,25 


.75 


.05 


4.92 


3.26 


15i 


15-16 


2.40 


.17 


8.38 


3.86 


.77 


.05 


2.66 


1.24 


m 


17-18 


2.45 


.18 


6.43 


2.35 


.78 


.06 


2.05 


.75 


20 


19-21 


2.50 


.19 


5.38 


1.62 


.80 


.06 


1.72 


.52 


23 


22-24 


2.55 


.21 


4.75 


1.23 


.81 


.07 


1.52 


.39 


27 


25-29 


2.61 


.23 


4.31 


.98 


.83 


.07 


1.36 


.31 


33 


30-36 


2.68 


.25 


3.97 


.79 


.85 


.08 


1.26 


.26 


42 


37^8 


2.75 


.27 


3.69 


.66 


.88 


.09 


1.8 


.21 


58 


49-72 


2.83 


.30 


3.49 


.57 


.90 


.10 


3.0 


.18 


97 . 


73-144 


2.93 


.33 


3.30 


.49 


.93 


.11 


1.5 


.15 


290 


145-rack. 


3.04 


.37 


3.18 


.42 


.97 


.12 


1.2 


' .13 



THE IMPROVED WILLIS ODONTOGRAPH. 

I have carefully calculated the distances nii n^ and nig n2 of the circles of 
centers from the pitch line, and also the radii aj m^ and a2 mg, and have 
arranged them in the table above, so that the data resulting from the usual 
process can be obtained without the usual labor. 

This improved Willis process will produce exactly the same circular arc 
as the usual method, with the same theoretical error, but its operation is 
simpler and less liable to errors of manipulation. 

By the usual process it is necessary to draw two radial lines, and to lay off 
a line at an angle with each. The tabular distances laid off on these lines, 
will locate the two centers. The two circles of centers are then drawn 
through them, and the dividers set to the radii to be used. 

By the new process the circles of centers are drawn at once without pre- 
liminary constructions, at the tabular distances from the pitch line, and the 
table also gives the radii to be taken on the dividers. No special instru- 
ment is required, no angles or special lines are drawn to locate the centers, 
and the chance of error is much less. 

This process, however, is not as correct, and is no simpler or more con- 
venient than the new odontographic process given further on. 




ROBINSON'S TEMPLET ODONTOGRAPH. 

This ingenious instrument, the invention of Prof. S. W. Robinson of the 
Ohio State University at Columbus, is based on the fact that some part of 
a certain curve of uniformly increasing curvature, called the logarithmic 
spiral, can be made to agree v^ith the true curve of a gear tooth with a degree 
of approximation that is very precise. 

It is a sheet metal templet having a graduated curved edge ac, shaped to 
a logarithmic spiral, and a hollow edge a b shaped to its evolute, an equal 
logarithmic spiral. 

To apply the instrument, draw a radial line from the pitch point d on 
the pitch line, and another from e, the center of the tooth, and then draw 
tangents d g and n e f , square with the radial lines. 

The instrument is then so placed that a certain graduation, given by 
accompanying tables, is at the point h on the tangent nef, while the grad- 
uated edge ac, is at the pitch point d, and the hollow edge ab, just touches 
the tangent line n e f at k, and then the face of the tooth is drawn with a pen 
along the graduated edge. The flank is similarly located by placing the 
instrument so that a certain other graduation is at the pitch point d, while 
its hollow edge touches the tangent line g d. 

The full theory of this instrument would be out of place here, but may be 
found in No. 24 of Yan Nostrand's Science Series, or in Yan Nostrand's Mag- 
azine for July, 1876. 



10 



A NEW ODONTOGRAPH. 



Having frequently to apply the "Willis Odontograph, it occurred to me 
that the process would be much simplified and much time and labor saved 
if the location of the circles of centers and the lengths of the radii were 
computed and tabulated, thus forming the improved Willis method already 
described. 

It was then evident that the process would be precisely the same, and the 
result much improved, if the centers tabulated were the centers of the near- 
est possible approximating circles, rather than of the Willis circles, and 1 
have embodied this idea in the following tables. 

I have carefully computed, by accurate trigonometrical methods, and have 
tabulated the location of the center of the circular arc that passes through 
the three most important points on the curve, at the pitch line a, fig. 0, 
at the addendum line k, and the point e, half way between. 

The tables locate this center directly, giving its distance from the pitch 
line, and from the pitch point. 

The circles of centers are drawn at the tabular distances "dis" inside and 
outside the jjitch lines, and all the faces and flanks are drawn from centers 
on these circles, with the dividers set to the tabular radii "rad." 

The tables are arranged in an equidistant series of twelve intervals. For 
ordinary purposes the tabular value for any interval can be used for any 
tooth in that interval, but for greater precision it is exact only for the 
given "exact" number, and intermediate values must be taken for inter- 
mediate teeth. 

The tables are arranged for both the diametral and circular pitch sys- 
tems. The former is much the more manageable and should be used when 
the work is not to interchange with work already made on the latter 
system. 

The first table, giving an interchangeable set, from twelve teeth upwards, 
is the one for general use. 

The second, or radial flank table, is inserted because teeth are sometimes 
drawn that way, but, as before explained, they a:ft weak, not interchange- 
able, and but a mere shade more direct in their action than the interchange- 
able style. 

ACCURACY OF THE ODONTOGRAPH. 

The assertion is often made that no circular arc can be made to do duty for 
the epicycloid, except for rough work, but it can be shown that the state- 
ment is not true if applied to the new method, for few mechanical processes 
can be made to work closer to a given example, than this arc is close to the 
true curve. 

Figure 9 shows the true curve, and both the 
new and the Willis aj)proximating arcs, the 
actual proportions being exagerated to show the 
errors more clearly. 

The Willis arc runs altogether within the true 
curve, while the new arc crosses it twice. 

We will take, for an example, the case of a 
twelve tooth pinion, which will show the errors 
at their greatest, and calculate them with great 
care for a tooth of three inch circular pitch, which 
is twice the size of the figure on page 13, and 
may be considered a very large tooth. 
The distance from pitch line to addendum line 
'''°- '• is divided into eight equal spaces by parallel cir- 

cles, and the distance along each circle, in ten thousandths of an inch, from 
the true curve to each odontographic arc, is as follows : 




11 





GRANT. 


WILLIS. 


At a 


.0000 


.0000 inches 


" b 


+.0088 


+.0175 " 


•' c 


+.0091 


+.0244 " 


" d 


+.0056 


+ .0283 " 


" e 


.0000 


+ .0288 '• 


" f 


-.0036 


+.0297 " 


u 


-.0061 


+.0308 " 


^' h 


-.0046 


+.0342 " 


" k 


.0000 


+.0397 " 



Average, .0042 .0260 " 

It is seen that the new arc is in no place one hundredth of an inch in error, 
and that for a tooth of four pitch, a large size for cut work, its average 
error is one thousandth of an inch. A greater accuracy than this would be 
of no practical value. 

The twelve tooth gear, for which the errors of both arcs were com- 
puted, shows them at their maximum value, for, as the number of teeth in 
the gear increases, the errors diminish, and for several locations their values 
for the new arc at c, which is the point of greatest error, are as follows : 



r t == 12 


= 


.009 inches 


" 20 


a 


.008 " 


" 40 


i. 


.006 '' 


" 100 


a 


.004 '' 


" 300 


a 


.002 '• 



and the errors of the Willis arc are subject to the same rule. 

The error of the Willis arc is plainly shown, at its greatest value, by the 
figure on page 13, where the dotted faces of the pinion teeth are correctly 
located by the Willis method. 

To further test the accuracy of the new method, construct the same tooth 
face several times by the same process, using either the method by points, 
or the usual Willis process. Unless the work is most carefully performed, 
it will be found that the several results will not agree with each other by 
amounts that are noticeable, while by the new method they will be sub- 
stantially the same curve. 

The new arc is most nearly correct at the most important point, the 
upper part of the curve, just where the Willis arc is most out of place, or 
where the true curve, unless drawn by some delicate and costly apparatus, 
in most likely to be out of place. 

CIRCULAR AND DIAMETRAL PITCHES COMPARED. 



CIR. P. 


DM. P. 


6 


.52 


5i 


.58 


5 


.63 


4i 


.70 


4 


.78 


3i 


.90 


3 


1.05 


21 


1.15 


2h 


1.25 


H 


1.40 


2 


1.57 


11 


1.80 


U 


2.10 


u 


2.50 


1 


3.14 


1 


4.20 


h 


6.28 



DM. P. 


CIR. P. 


h 


6.28 


1 


4.20 


1 


3.14 


u 


2 50 


u 


2.10 


11 


180 


2 


1.57 


2h 


1.25 


3 


1 05 


3i 


.90 


4 


.78 


6 


.63 


6 


.52 


7 


.45 


8 


.39 


9 


.35 


10 


.31 



12 



THE NEW ODONTOGRAPH. 










GENERAL DIRECTIONS. 

Draw the pitch line and divide it for the pitch points mag. Take from 
the tables, multiply or divide, as the case may require, by the pitch in use, 
and lay olf , the addendum a b and a c, the clearance e f , the backlash g- g', 
the face distance a d, and the flank distance a c. Draw the addendum line 
through b, the root line through e, the clearance line through f, the line 
of face centers through d, and the line of flank centers through c. Set the 
dividers to the face xadius, and draw all the faces ab from centers A. Set 
to the flank radius, and draw all the flanks a k from centers B. Round the 
flanks into the clearance line. The flanks of a gear of twelve teeth are 
straight radial lines. 

ODONTOGRAPH TABLE. 
EPICYCLOIDAL TEETH. 

INTERCHANGEABLE SERIES. 
From a Pinion of Twelve Teeth to a Rack. 







FOR ONE 


FOR ONE INCH J 


NUMBER OF 

TEETH 


DIAm 

For ar 


[ETRAL P 


ETCH. 

dde hy 


CIB 

Fors 


.CUI.AR PITCH. 1 


ly other pitch, di 


my other pitch, multiply 1 


IN THE GEAR. 


that pitch. 1 


hy that pitch. | 






Faces. 


Flanks. 


Faces. 


Flanks. 1 


Exact. 


Intervals. 


Rad. 


Dis. 


Rad. 


Dis. 


Rad. 


Dis. 


Rad. 


Dis. 


12 


12 


2.01 


.06 


CO 


CO 


.64 


.02 


CO 


oo 


134 


13-U 


2.04 


.07 


15.10 


9.43 


.65 


.02 


4.80 


3.00 


15i 


15-16 


2.10 


.09 


7.86 


3.46 


.67 


.03 


2.50 


1.10 


17i 


17-18 


2.14 


.11 


6.18 


2.20 


.68 


.04 


1.95 


.70 


20 


19-21 


2.20 


.13 


5.12 


1.57 


.70 


.04 


1.63 


.50 


23 


22-24 


2.26 


.15 


4.50 


1.13 


.72 


.05 


1.43 


.36 


27 


25-29 


2.33 


.16 


4.10 


.96 


.74 


.05 


1.30 


.29 


33 


30-36 


2.40 


.19 


3.80 


.72 


.76 


.06 


1.20 


.23 


42 


37-48 


2 48 


.22 


3.52 


.63 


.79 


.07 


1.12 


.20 


58 


49-72 


2 60 


.25 


3 33 


.54 


.83 


.08 


1.06 


.17 


97 


73-144 


2.83 


.28 


3.14 


.44 


.90 


.09 


1.00 


.14 


290 


145-rack. 


2.92 


.31 


3.00 


.38 


.93 


.10 


.95 


.12 



13 



A PRACTICAL EXAMPLE 



OF THE WORK OF THE NEW ODONTOGRAPH, 




face dis, = .07. 
flank " = .54. 
face " = .03. 
flank " r= CO 



INTERCHANGEABLE SERIES. 

Example. — A gear of 24 teeth, and a gear of 12 teeth, of H circular 
pitch. 

Data. — Take from the table the numbers to be used, which are as follows 
when multiplied by 1^. 

For 24 teeth, face rad, = 1.08 
" 24 '' flank " =2.15 
'' 12 " face " — .96 
" 12 " flank '' = cc 

Also take from the proper tables the pitch diameters 5.73 and 11.46 inches, 
the addendum, .5 inch, and clearance, .06 inch. 

Pkocess. — Draw the two pitch lines, and divide for the pitch points. Draw 
the addendum, root, and clearance lines of both gears. 

Draw the circles of centers, .03 inside the pitch line of the 12 tooth gear, 
and .07 inside of that of the other. Draw the circles of flank centers, .54 
outside the pitch line of the 24 tooth gear, and draw straight radial flanks 
for the 12 tooth gear. 

Draw the faces of the 12 tooth gear with the radius. 96, and draw the faces 
of the 24 tooth gear with the radius, 1.08, and the flanks with the radius 2.15. 

Kound the flanks into the root line, and allow backlash by thinning the 
teeth according to judgement. 

The dotted faces of the 12 tooth gear show them as they would be laid 
out by the Willis odontograph, and the figure also shows the two centers in 
place. 



14 



RADIAL FLANK SYSTEM. 

TEETH NOT INTERCHANGEABLE. 

Gears on this system must work together in pairs, each gear being fitted to 
its mate and to no otner. See page 3. The process is the same that has 
been described on page 12 for the interchangeable set. 




RADIAL FLANK SYSTEM. 

ExPLANATioisr OF THE TABLE. — The Upper number in each square is tlie 
face radius, the lower is the center distance. 

The centers are mostly insid the pitch line, but some are on the line, and 
those having the negative sign are outside of it. 

The tabular numbers are for one inch circular pitch, and must be multi- 
plied by any other circular pitch in use. For the value for any diametral 
pitch, multiply the tabular number by 3.14, and then divide by the diame- 
tral pitch in use. 

Example. — A gear of 12 teeth, paired with a gear of 24 teeth. Circular 
pitch-l^ inches. 

Data. — Take from the table for 12 teeth into 24, face radius =.68 and cen- 
ter distance = 0, and for 24 teeth into 12. radius == 72, and distance = .05. 
These multiplied by 1^ give the values for use on the drawing, 12 rad. =1.02, 
12 dis = 0, 24 rad. = 1.08, and 24 dis. = .07. 

The addendum is one third the pitch, = i inch, and the proper tables give 
the clearance =.06, and the pitch diameters = 5.73 and 11.46 inches. 

Process, — Draw the two pitch linos 5.73 and 11.46 inches in diameter and 
space them for the teeth. 

Lay off the addendum, .5 inch, and the clearance, .06 inch, and draw the 
addendum, root, and clearance lines. 

Draw all the faces of the twelve tooth gear, from centers on its pitch line, 
with the radius 1.02. Draw all the faces of the 24 tooth gear from centers 
on a line .07 inch inside its pitch line, with the radius 1.08 inches. Draw 
straight radial lines for the flanks of all tlie teeth. 



15 



ODONTOGRAPH TABLE. 

EPICYCIiOIDAIi TEETH. 

RADIAL FLANK TABLE. 

FOR ANY POSSIBLE PAIR OF GEARS, NOT INTERCHANGEABLE. 

One Inch Circitlar Pitch. 
For any other pitch, multiply by that pitch. 



Ntm 

TEETH 
BEING 

Exact. 


BER OF 
JN GEAR 
DRAWN. 

Intervals 


NUMBER OF TEETH IN THE MATE. 

.„ 13 15 17 19 22 25 30 37 49 73 145 
^" 11 16 18 21 24 29 36 48 72 144 rack 


12 


12 


.64 
.02 

.65 
.02 


.64 
.01 


.65 
.01 


.66 
.01 


.67 



.68 



.69 
-.01 


.70 
-.01 


.71 
-.02 


.73 
-.02 


.74 
-.03 


.75 
-.03 


ISh 


13-14 


.66 
.02 


.67 
.Oi 


.68 I .69 
.01 .01 


.70 



.72 



.74 
-.01 


.75 
-.01 


.76 
-.02 


.78 
-.02 


.79 
-.03 


15i 


15-16 


.67 
.03 


.68 
.02 


.69 
.02 


1 
.70 1 .72 
.01 .01 


.74 
.01 


.75 



! .78 



.79 
-.01 


.82 
-.02 


.84 
-.02 


.84 
-.03 


m 


17-18 


.68 
.04 


.70 
.03 


.71 
.02 


.73 
.02 


.75 
.01 


.77 
.01 


.78 
.01 


.82 



.84 
-.01 


.87 
-.01 


.89 
-.02 


.90 
-.03 


20 


19-21 


.70 
.04 


.72 
.04 


.74 
.03 


.76 
.02 


.79 
.02 


.81 
.01 


.83 
.01 


.87 



.90 



.93 
-.01 


.96 
-.02 


.96 

-.03 


23 


22-24 


.72 
.05 


.74 
.04 


.76 
.04 


.79 
.03 


.82 
.02 

.85 
.03 


.84 
.02 


.87 
.01 


.91 
.01 


.94 



.98 
-.01 


1.01 
-.02 


1.03 
-.03 


27 ' 23-29 


.74 
.05 


.76 
.05 


.79 
.04 


.82 
.04 


.87 
.02 


.92 
.02 


.96 
.01 


.99 



1.03 
-.01 


1.07 
-.02 


1.10 
-.03 


33 1 30-36 

! 


.76 
.06 


.79 
.05 


.83 
.05 


.86 
.04 


.90 
.03 


.94 
.03 


.98 
.02 


1.02 
.01 


1.06 



1.11 



1.17 
-.01 


1.23 
-.02 


42 


37-48 


.79 
.07 


.83 
.06 


.86 
.05 


.90 
.05 


.96 
.04 


.98 
.04 


1.03 
.03 


1.08 1.14 
.03 .02 


1.20 



1.25 



1.37 
-.01 


58 


49-72 


.83 
.08 


.87 
.07 


.91 
.07 


.96 

.06 


1.02 
.06 


1.05 
.05 


1.10 
.04 


1.17 1.24 
.04: .03 


1.30 
.02 


1.43 i 1.58 



! 

97 


73-144 


.90 
.09 


.93 

.08 


.97 
.08 


1.01 
.07 


1.07 
.07 


1.11 
.06 


1.18 
.06 


1.28 1.34 
.05 .04 


1.47 
.03 


1.65 
.02 


2.03 



290 145 rack 


.93 
.10 


.96 
.09 


1.00 
.09 


1.05 
.09 


1.10 

.08 


1.16 

.08 


1.24 
.07 


1.37 ' 1 50 
.07 1 .06 


1.70 
.04 


2.12 
.03 


2.90 
.02 



16 



THE INVOLUTE TOOTH. 




With the exception of the epicycloid, the only curve in extensive use for 
the working face of a gear tooth, is the involute. 

THE INVOLUTE CURVE. 

As the rolling circle A of fig. 3 increases in size, it finally, when of infinite 

diameter, becomes the straight line d g of fig. 
15, while the epicycloid traced by a fixed point 
in the circle becomes the involute. 

The involute is, therefore, not a new or sep- 
arate curve, but simply a particular case of the 
epicycloid. It is the infinite form of the epicy- 
cloid. 

As the rolling circle of infinite diameter is the 
same thing as a straight line, the involute can 
be formed by a fixed tracing point in a cord 
which is unwound from a circle, called its "base 
circle," which has been wrapped or " involved" 
FIG. 15. THE INVOLUTE. 1^ It, aud from this property it derives its name. 

ITS UNIFORM ACTION. 

If the two circles A and B, fig. 16, are separ- 
ated by the distance ab, and work together by 
means of two external epicycloids G and D, the 
motion communicated will be irregular, for the 
conditions of uniformity are that the two cir- 
cles shall touch, and that the external curve 
of one shall work with the internal curve of the 
other. See page 2 and figure 4. 

The amount of this irregularity will depend 
on the proportion between the separating dis- 
tance ab and the diameter of the rolling circle 
which describes the epicycloids. If the pro- 
portion is very small, the irregularity will be 
very small, and if the rolling circle has an in- 
finitely great diameter, the proportion and the 
irregularity will be infinitely small, that is, zero. Therefore, involutes 
will work together with perfect regularity and are suitable curves for gear 
teeth. 

ITS ADJUSTIBILITY. 

If tlie rolling circle is infinitely large, the proportion between the separat- 
ing distance and it will always be zero, and it will not be altered by any finite 
alteration of the former, and therefore the uniformity of the action of 
involute teeth is not in any way dependent upon, or affected by any change of 
the separating distance. The action will be perfect as long as the curves 
remain in contact, and this is a property of the greatest practical value, 
which gives the involute a great advantage over every other known or pos- 
sible curve. 

The curve of any gear tooth must of necessity be a "rolled curve " formed 
by a fixed object attached to the plane of or moving with some curve that 
rolls upon the base curve of the tooth, and, as the involute is the infinite 
form of any rolled curve, it is the only form that can possess this property 
of adjustibility. 




EXTERNAL EPICYCLOIDS. 



17 



ITS UNIFORM PRESSURE AND FRICTION. 

The point of contact oi the two involutes C and D will always be upon the 
sti'ai<?ht line of action mn, the common tangent of the two base circles, 
commencing at its point of taugeucy with one circle, and ending at the same 
point on the other. 

The direct pressure between the two teeth will always be in the direction 
of the line of action, and uniform both in direction and in amount, a prop- 
erty that is peculiar to the involute curve, and which contributes greatly to 
the smooth action and even wear of involute teeth. Friction is substan- 
tially in proportion to direct pressure, and when the pressure is uniform, 
the friction will be uniform, and no part of the curve w^ill be more likely to 
wear away than any other, part. The durability of a tooth, particularly 
when doing heavy work, depends on the uniformity of the friction as well 
as upon its absolute amount. 

THEORETICAL CONSTRUCTION. 

To draw the involute curve through the pitch point a of two pitch circles 
A and B, draw^ the line of action m n at any desired angle w^th the line of 

centers, usually 75°, and then draw 
the base circles C and D, touching the 
line of action at e and d, where the 
perpendicular radial lines e g and f d 
meet it. From a, step off anj^ num- 
ber of short steps along the line of 
action and around the base line to any 
point s, then draw any number of 
tangent lines b c, t v, then step off the 
distances sbc, stv, sb, etc., each 
equal to s d a, and the jDoints c, v, b, 
etc., will be points of the curve. Any 
line, as w c X drawn through c at right 
angles to he, will be tangent to the 
curve. The working part of the 
curve must not be extended beyond 
the circle k e p through the point of 
contact of the line of action m n and 
the base line C, for beyond that point it will interfere with the radial flank 
of the tooth it works with. 

The curve is generally limited by the addendum line z y, at an arbitrary 
distance from the pitch line B, and ends at b on the base line D, where it is 
perpendicular to the base line. It is continued within the base line by a 
radial line as far as the root line z y, and is then rounded into the clearance 
line. 

The matter under epicycloidal teeth, pages 3, 4, and 5, regarding the pitch, 
addendum, clearance, and backlash, will apply as well to involute teeth. 

ANGLE OF ACTION. 

The angle m a g may be less, but not greater, than the value found from 
the formula 

180° 
mag = 90° — -—- 

in which s is the number of teeth in the smallest gear in the pair. If the 
angle is greater than this the motion will not be continuous, as each pinion 
tooth will pass out of action before the next one is in position to act. 

INTERCHANGEABLE SETS. 

Any number of involute gears from base circles of different diameters will 
work together correctly and interchangeably if all are of the same pitch, and 
have the same angle of action. 

If we put s = 12 teeth, we find 

m ag = 90*^ — ^2~ = ^^ 




CONSTRUCTION. 



18 




FIO. 18. THE OLD RULE. 



the value for the common twelve to rack interchangeable set, and if we use 
fifteen as the smallest number of teeth in the set, we have an angle of action 

of 78°. 

PRACTICAL CONSTRUCTION. 

When the involute is to be brought into use, we meet with the same diflS- 
culties as with the epicycloid, for its theoretically correct construction is 
not easily and accurately accomplished, and we must adopt some short cut 
of approximative accuracy. 

The principle of the epicycloidal engine of fig. 7 may be applied to the 
construction of the involute, the ribbon s being drawn tight and straight as 
it is unwound from th'e base circle, but the same difficulties prevent its use 
for ordinary purposes. 

THE OLD RULE. 

A defective rule in common use draws the whole curve from base line to 

addendum line, as one circular arc. The angle 
mag is laid off at 75°, sometimes at 75^°, the 
distance a c is made equal to one quarter of the 
pitch radius a g, and the tooth curve is drawn 
from c as a center. 

This rule is simple, to be sure, but it gives the 
faces shown by the dotted lines of the figure on 
page 22, and is abominably wrong and worth- 
less. 

If it would round off the points of the teeth 
of a large gear, it would be useful to correct 
interference, but it greatly rounds the teeth of 
a small gear that needs little or no correction, 
and gives the curve on a large gear in nearly its 
theoretical position, without the allowance for 
interference that must be made. 

It is not to be wondered that the involute tooth is in small favor with 
practical mechanics who use this bungling method, and who do not under- 
stand that the trouble is not in the involute system, but in its defective 
application. 

A NEW METHOD. 

In devising a method for drafting the involute tooth, I have borne in 

mind that a minute degree of accu- 
racy is not the essential requirement, 
for although substantial accuracy 
must be secured, simplicity and con- 
venience are qualities that must also 
be considered. 

The method, in general terms, and 
given in full on pages 22 and 23, is to 
give, by a table, the distance of the 
base circle B, see fig. 19, inside the 
pitch circle P, and to give by the same 
table, tlie distances or radii a c and a d 
from the pitch point a to centers c and 
d on the base line. The face arc aw is 

drawn from the center d and the flank arc a v from the center c. 
The table, page 23, is for one diametral pitch, and covers the common 

twelve to rack interchangeable set. Interference must be corrected, when 

necessary, as explained below in detail. 

INTERFERENCE. 

As indicated above, the involute face will interfere with the radial flank 
of the mating tooth if the addendum is greater than a certain amount, and 
as the addendum in common use for the interchangeable set generally 
exceeds this limit, we must gererally make corrections to avoid this trouble. 




FIG. 19. THE NEW METHOD. 



19 




FIG. 20. INTERFERENCE. 



Interference Table 

For one diametral pitch and 3 1-7 inch cir- 
cular pitch. Angle of action, 75°. 

Number of Teeth in the Mate. 
13 15 17 19 
12 14 16 18 21 



Figure 20 shows the interference, its effect, and its correction. 

The working face of the involute 
should be Umited at i by the circle 
k p through the tangent point e, but 
if the usual addendum continues it 
beyond that line, to s, the extension 
si will interfere with the radial flank 
c f , and the uniformity of the action 
, will be destroyed. 
'" To correct it we must either 
"weaken and spoil the shape of the 
mate tooth by undercutting the 
flank c f by an epitrochoidal line c g, 
or we may, and much better, round 
off the point of the tooth by an epi- 
cycloidal curve i h. 
The amount of this interference will depend on, and increases with, the 
angle of action, and also depends upon the number of teeth in each gear. It 
is greatest on a large gear or rack that runs in a small pinion, and least on 
a pinion running in a large gear. When the angle of action is 75° there 
is no interference when both gears of a pair have thirty or more teeth, or 

when an equal pair have twenty-one or 
more teeth. When on3 gear has more, 
and the other has less than thirty teeth, 
the larger may need correction, but the 
smaller never will. 

The amount of the interference, the 
correction to be made by rounding off the 
point of the tooth, is very small and may 
generally be neglected on small pinions. 
It is given by the lower figures in the 
table, which shows that it is never more 
than a sixteenth of an inch on a large 
tooth of one diametral, or three inch 
circular pitch, and not over two or tliree 
hundreths of an inch on a gear of that 
pitch having few teeth. The table also 
shows by the upper figures the limit 
point or distance ix above the pitch line 
where the interference commences. 

The tabular numbers must be divided 
by the diametral pitch that may be in use, 
and for any circular pitch it is sufficient 
to divide the tabular number by 3 and 
then multiply by the pitch. 

The table takes no notice of an inter- 
ference of less than a hundredth of an inch 
on a tooth of three inch circular pitch. 

When, as is usually and should always 
be the case, the gear being drawn belongs 
to the twelve to rack interchangeable set, 
the interference should be computed for 
a mate gear of twelve teeth, or by the first 
vertical column of the table. In this case 
the error will not be perceptible if the 
limit distance to point of first interfer- 
ence be always assumed to be half the 
addendum. * 

When the work is upon a rough cog 
wheel or mill gear, or upon a pattern for 
a cast gear, the only correction needed 
for interference, is a slight rounding off 
of the points if it is a rack or very large 
gear, and a mere touch on the point of a 
gear of few teeth. 



Q 

bo 

c 

i- 

a 

0) 

O 

0) 



12 


.58 
.01 


.67 
.01 








13-14 


.56 
.02 


.66 
.01 








15-16 


.54 

.02 

.53 
.02 


.64 
.01 


.75 
.01 






17-18 


.62 
.02 


.72 
.01 


19-21- 


.51 
.02 


.60 
.02 


.69 
.01 






22-24 


.50 
.02 


.58 
.02 


.67 
.01 






25-29 


.40 
.03 


.57 
.02 


.65 
.02 


.75 
.01 




30-36 


.47 

.03 


..55 
.02 


.63 
.02 


.72 
.01 




37-48 


.45 

.03 


.53 

.02 


.61 
.02 


.69 
.01 




49-72 


.44 
.04 


.52 
.03 


.59 
.02 


.66 
.02 


.73 
.01 


73-144 


.42 
.05 


.49 
.04 


..56 
.03 


.63 
.02 


.70 
.01 


145-00 


.40 

.06 


.46 

.05 


.53 
.04 


.60 
.02 


.67 
.01 



20 
EPICYCLOIDAL vs. INVOLUTE TEETH. 

A COMPARISON. 

The epicycloidal tooth is in much greater use and favor than the involute 
form, particularly for heavy work, both writers and mechanics generally 
preferring it, and seldom giving the preference to its rival. It is difhcult to 
account for this favor except, as in the case of the circular pitch system, on 
the ground that the epicycloid was adopted in the infancy of mechanical 
science, and holds its place by virtue of prior possession, for the involute 
has certainly the advantage from every practical point of view. 

Space will not i^ermit an extended discussion with the necessarily bulky 
demonstrations, but, if the two curves be closely and carefully examined 
under the same conditions within the limits of either the twelve tooth or the 
fifteen or higher tooth interchangeable series, with the customary adden- 
dum, which limitation will cover nine-tenths of tlia gears in actual use, it 
will be found that they compare as f oIIoavs : 

I. Adjustibility. Involute teeth alone can possess the remarkable and 
practically invaluable property, that they are not confined to any fixed 
radial position with respect to each other, for, as long as one pair of teeth 
remains in action mitil the next pair is in x^osition, the perfect uniformity 
of the action of the curve is not impaired. 

The shafts may be at the proper distance apart, or not, as happens, and 
they may change position by wearing, or variably as when used on rolls, or 
may be forced together to abolish backlash, and, in fact, the curve is won- 
derfully adapted to the variable demands, and will accommodate itself to 
errors and defects that cannot be avoided in practice. 

Epicycloidal teeth must be put exactly in place and kept there, and the 
least variation in position, from bad workmanship in mounting, or by wear 
or alteration of the bearings in use, will destroy the uniformity of the 
motion they transmit. When perfectly mounted and carefully kept in 
order, epicycloidal teeth are as good as any in this respect, but for most 
practical purposes they are decidedly inferior. 

This virtue of the involute is always recognized by writers, but is seldom 
given the position its importance demands, for it is only as a result of expe- 
rience in making and using gears, that its importance can be seen at its full 
value. 

II. Uniformity. The direct force exerted by involute teeth on each 
other, is exactly uniform, both in direction and in amount, and this property 
ensures a uniform wearing action of the teeth, a nearly uniform thrust on 
the shaft bearings, and a steadiness and smoothness of action that cannot be 
claimed for epicycloidal teeth under any circumstances. 

The direct pressure acting between epicycloidal teeth is variable in 
amount and very variable in direction, and consequently the friction and 
wearing action between the teeth, as well as the thrust on the bearings, is 
variable between wide limits. 

III. Friction. The measure, for purposes of comparison, of the loss of 
power by friction, is the product of the direct pressure between the teeth, 
multiplied by their rate of sliding motion on each other. 

This measure is always in favor of the involute by a decided advantage, 
although the advantage is usually claimed for the epicycloid, both as to 
maximvim values and average values, and as this is an important point, it 
should have great weight in deciding between the two forms of teeth, for 
the element of friction is of chief importance in determining the life of a 
gear in continual and heavy service. 

The epicycloid is mostly in use for heavy gearing from a mistaken view of 
this point, it being generally taught that its friction is the least. 

IV. Thbhtst on Bearings. Here the advantage is with the epicycloidal 
tooth, but not by a large amount, and not in a matter of first consequence. 

The thrust on the bearings due to the action of the teeth on each. other is 
but a fraction of the whole thrust due to the power being carried, and as 



21 



the average thrust of the teeth is but little in favor of the epicycloid, and as 
the maximum thrust is always from that form of tooth, the two forms may 
be said to be well balanced in this respect. Moreover, the thrust of the 
involute is but slightly variable, \vhile that of the epicycloid varies from 
large values at the points of first and final action to nothing at all at the 
line of centers, and must give rise to a rattling and uneven action. 

Y. Strength. The weakest part of a tooth is at its root, and as the 
involute tooth spreads more than the epicycloidal tooth, it is stronger at that 
point and has a considerable advantage. 

VI. Appearance. This is a small point and a matter of opinion, but 
is worth mention. The involute is a simple and graceful single curve, while 
the epicycloid is a double and not mechanically a neat curve, and, as gener- 
ally drawn, has a decided bulge or even a plain corner w^here the two halves 
join at the pitch line. 

In General. As the involute has the advantage of the epicycloid, in 
nine actual cases out of ten, with respect to adjustibility in position, in 
uniformity of wear and action, in loss of power and change of shape by 
friction, in strength, and in appearance, and is but a shade, if any, inferior 
with regard to the thrust on the bearings, it may be, and should be accorded 
first place for any and every practical purpose. The writer can imagine no 
possible case, unless it be in connection with a pinion of very few teeth, 
where the epicycloid would have either a theoretical or a practical advan- 
tage over the involute. 



INVOLUTE TEETH 

OF SEVERAL DIAMETRAL PITCHES. 




22 



ODONTOGRAPH TABLE. 



INVOLUTE TEETH. 

INTERCHANGEABLE SERIES. 
From a Pinion of Twelve Teeth to a Rack. 









rOR ONE 




FOR ONE INCH 


NUMBER OP 

TEETH 

IN THE 6EAK. 


DIAMETKAI. PITCH. | 


CIRCULAR PITCH. 


For any 


other pitch, 
that pitch. 


divide by 


For any 


other pitch, multiply 1 
by that pitch. 1 


Exact. 


Intervals. 


Base 
Distance. 


Face 
Radius. 


Flank 
Radius. 


Base 
Distance. 


Face 
Radius. 


Flank 
Radius. 


12 


12 


.20 


2.70 


.83 


.06 


.86 


,27 


13 


13 


.22 


287 


.93 


.07 


.91 


.30 


14 


14 


.23 


3.00 


1.02 


.07 


.95 


.33 


15 


16 


.25 


3.15 


1.12 


.08 


1.00 


.36 


16 


16 


.27 


3.29 


1.22 


.08 


1.05 


.40 


17 


17 


.28 


3.45 


1.31 


.09 


1.09 


.43 


18 


18 


.30 


3.59 


1.41 


.09 


1.14 


.46 


19 


19 


.32 


3 71 


1.53 


.10 


1.18 


.50 


20 


20 


.33 


3.86 


1.62 


.10 


1.22 


.53 


21 


21 


.35 


4.00 


1.7;^ 


.11 


1.27 


.57 


22 


22 


.37 


4 14 


1.83 


.11 


1.32 


.60 


23 


23 


.39 


4,27 


1.94 


.12 


1.36 


.63 


25 


24- 26 


.42 


4.56 


2.15 


.13 


1.45 


.70 


28 


27- 29 


.45 


4.82 


2.37 


.14 


1.54 


.77 


31 


30- 32 


.50 


5.23 


2.69 


.15 


1.67 


.88 


34 


33- 36 


.57 


5 77 


3.13 


.17 


1.84 


1.00 


38 


37- 41 


.63 


6 30 


3.f.8 


.19 


2.01 


1.^6 


44 


42- 48 


.73 


7.08 


4.27 


.22 


2 26 


1 38 


52 


49- 58 


.87 


8.13 


5.20 


.26 


2.69 


1.70 


64 


59- 72 


1.07 


9.68 


6.G4 


.32 


3.09 


2.18 


83 


73- 06 


1.39 


12.11 


8.93 


.42 


3.87 


2.90 


115 


97-144 


1.92 


1G.18 


12 80 


.58 


5.16 


4.15 


192 


14.">- 288 


3.20 


2;-.. 86 


22.30 


.96 


8.26 


7.3> 


576 


289-rack 


9.60 


73.95 


70.10 


2.88 


23.65 


22.30 



INTERFERENCE 
FOR TWELVE TO RACK INTERCHANGEABLE SET. 



Teeth 
In the gear. 


12 


13 
14 


15 17 

16 18 


19 
21 


22 
24 


25 
29 


30 
36 


37 
48 


49 1 73 
72 1 144 


145 

00 


Pitch. 






Amount of the Interference. 






One in. cir. 


.003 


.007 


.007 .007 


.007 


.007 


.010 


.010 


.010 


.013 .017 


.020 


One diamet'll 


.01 


.02 


.02 .02 


.02 


.02 


.03 


.03 


.03 


.04' .05 


.00 



Interference always to commence at a point half way between pitch 
line and addendum line. 



23 



A PRACTICAL EXAMPLE. 




INVOLUTE TEETH. 

INTERCHANGEABLE SERIES. 

Example. — A rack, and a pinion of twelve teeth, of two diametral pitch. 

Data. — From the tables we have, after dividing by 2, the base distance 
.10'', face radius 1.35", and flank radius .42'^ The addendum is .5", the 
clearance .06", and the pitch diameter 6. inches. The limit of interference 
for the rack is .20", and for the gear .28", or may be assumed at .25" with 
small error. The amount of interference for the rack is .03", and for the 
gear .005". 

PiiOCESS. — Draw the pitch lines and divide for the pitch points, lay oft 
the addendum and the clearance, and draw the addendum, root, and clear- 
ance lines. Draw the base line .10" inside the pitch line of the gear. Draw 
the limit lines .20" and .28" from the pitch lines. With the face radius 3.35" 
and from the center d on the base line, draw the faces of the gear from pitch 
line to limit line, and, as the interference is imperceptible in this case, con- 
tinue it to the addendum line. With the flank radius .42" and from the center 
b on the base line, draw the flank from pitch line to base line. The flank, 
inside the base line, is a straight radial line. 

The face and flank of the rack is a straight line from root line to limit 
line, at an angle of 75° with the pitch line. From the limit line to the 
addendum linc^ the face of the rack curves inward, being .03 inch from the 
true lace at the point. 

The interference on the pinion tooth is here neglected, because it is very 
small, but on larger gears it must be accounted for if the gear is to belong- 
to the interchangeable set. 

At sixty teeth the root and base lines coincide, and there is no radial 
flank. 



24 




BEVEL GEARS. 

In laying out the teeth of a bevel gear but one new point needs to be con- 
sidered. The working pitch diameter a b c is not to be used, but the teeth 
are to be drawn on the conical pitch diameter adc, developed or rolled out 
as in fig. 25. 

The conical diameter adc may be found from a drawing, or if the gears 
are of some common proportion, from the following table by multiplying 
the true pitch diameters by the tabular numbers given for that proportion 

TABLE OF CONICi^L PITCH DIAMETERS 
OF BEVEL GEARS. 



Proportion. 


Larger Gear. 

1.41 


Smaller Gear. 


1 to 1 


1.41 


2 » 1 


2.24 


1.12 


3 " 2 


1.80 


1.20 


3 " 1 


3.16 


1.05 


4 " 3 


1.67 


1.25 


4 " 1 


4.12 


1.03 


5 " 4 


1.60 


1.28 


5 " 3 


1.94 


1.17 


5 " 2 


2.69 


1.08 


5 " 1 


5.10 


1.02 


6 " 5 


1.56 


1.30 


6 « 1 


6.08 


1.01 


7 " 1 


7.07 


1.01 


8 " 1 


8.06 


1.01 


9 " 1 


9.06 


1.01 


10 '' 1 


10.05 


1.01 



Examples. — A miter gear, proportion 1 to 1, of 4 pitch, 6" diameter, and 
24 teeth, has a conical diameter of 6'^ x 1.41 = 8.46'^ and there are 24 x 1.41 
= 33.8 teeth on the full circle of the developed cone. 

A pair of bevel gears of 3 to 1 proportion, 48^' and W diameters, 36 and 12 
teeth, have conical diameters 48'' x 3.16 — 151.68". and 16" x'5rt6 = 16.80", 
and there are 36 x 3.16 — 113.76, and 12 x 1.05 = 12.60 teeth on thje full cir- 
cles of the developed cones. 



\^S 



25 




INTERNAL GEARS. 

The internal gear, sometimes called the "annular" gear, is drawn by the 
rules for spur gears, the teeth of a spur gear being the spaces between the 
teeth of an internal gear of the same pitch diameter, with the backlash and 
clearance reversed in position. 

Involute teeth should end at the base line, the radial part of the flank 
being omitted, or well rounded over if it is desirable to preserve the appear- 
ance of the full tooth. 

Internal teeth will interfere, even if properly drawn, unless the gear is 
considerably larger than the pinion running in it. If drawn for the common 
twelve to rack interchangeable set, there should be at least twelve more 
teeth in the gear than in the pinion, and if the difference is less, the teeth 
must be "doctored " or rounded over until they will pass, and there must 
be a difference of two teeth in any case. 

Involute teeth have a decided advantage over epicycloidal teeth for inter- 
nal gearing, their action being much more direct, with less sliding and 
friction. 



26 
STRENGTH AND HORSE-POWER OF GEARS. 



There are about as many different rules for this purpose, and contradictory re- 
sults, as there are writers upon the subject. I have preferred not to discuss the 
theory, but to adopt without quesiion the method given by Thomas Box in liis Prac- 
tical Treatise on Mill Gearing, because that engineer has most carefully considered 
the practical points in view, and because his formulze agree almost exactly with a 
great mauy cases in actual practice. 

STRENGTH OP A TOOTH. —For worm gears, crane gears, and slow-moving 
gears in general, we have to consider only the dead weight that the tooth can lift 
with safety. 

If we allow the iron to be subjected to but one tenth of its breaking strain, we 
can use the formula: — 

W = 350 c f, 

in which "W is the dead weight to be lifted, c is the circular pitch, and f the face, 
both in inches. 

For the wooden cogs of mortise wheels, use 120 instead of 350 as a factor in the 
formula. 

When the pinion is large enough to insure that two teeth shall always be in fair 
contact, the load, as found by this rule, may be doubled. 

Example. — A cast-iron gear of 3" circular pitch and G" face will lift 
W = 350 X 3 X 6 = 6300 lbs. 



HORSE-POWER OF A GEAR. — For very low speeds we can use the 

formula, 

HP for low speed = .0037 d n c f , 

in w^hich d is the pitch diameter, c the circular pitch, and f the face, all in inches, 

and n is the number of revolutions per minute. 

Example. — The horse-power of a gear of three feet diameter, three inch 

pitch, and ten inch face, at eight revolutions per minute, is, 

HP = .0037 X 36 X 8 X 3 X 10 = 32. 



For ordinary or high speeds, where impact has to be considered, it is found that 
the above formula gives too high results, and we must use the formula, 

HP at ordinary speeds = .012 c^ f \/du. 

Example. - A gear of three feet diameter, three inch pitch and ten inch face, 
at one hundred revolutions per minute, will carry but 

HP == .012 X 9 X 10' X VlOO X 36 = 65 horse-power, 
instead of the 400 horse-power found by the rule for low speeds. 



, At ordinary or high speeds a wooden cog, on account of its elasticity, will carry 
as much as or more power than a cast-iron tooth, and we can use .014 instead of .012 
in the formula. 

"When in doubt as to whether a given speed is to be considered high or low, com- 
pute the horse-power by both formulae, and use the smallest result. 



For bevel gears the same rules will apply, if we use the pitch diameter and the 
pitch at the center of the face. 

Some ru!es in use take no account of the face of the gear, but assume that the 
tooth should be able to bear the whole strain upon one corner. 

A tooth that does not bear substantially along its whole face, at several points at 
least, is a very poor piece of work, and it would be better to straighten the tooih 
than to force the rule to follow it. 



27 



THE EQUIDISTANT SERIES. 

The shape of a tooth is not the same on two gears of different sizes, for 
its curvature continually decreases and the curve flattens as the number of 
teeth in the gear increases. 

When the teeth are formed by a rotary milling tool, we must use a cutter 
of fixed shape ; when formed by planing, a fixed guide is employed ; and 
when drawn by an odontograph, flxed tabular data are used ; and obviously, 
if we require the greatest possible accuracy we must have a different shape 
of cutter, or guide, or a separate tabular number, for each separate tooth, 
and at least two hundred in a set to cover the ordinary range of work. As 
this would be an expensive and clumsy system, it is customary to make one 
fixed shape do duty for several teeth, being just right for one tooth of a 
given interval, and approximately so for several teeth either way. 

This set of fixed intervals is known as the equidistant series, as it so dis- 
tributes the errors that the greatest error is the same in all the intervals. 

The equidistant series was invented by Willis, but he gives no rule for 
arranging it, and the example he gives was apparently found by some 
experimental method. 

In the American Machinist for Jan. 8th, 1881, I proposed the location of 
the dividing points of the series by the formula 



, a n 

n-s-4- — 

I 7. 



in which a is the first and z the last tooth, usually twelve and infinity, of a 
series of n intervals, s is the number, in the series, of any particular inter- 
val, and t is the last tooth in the interval s. 

This formula uniformly distributes, not the differences in form, but what 
is for all i^ractical purposes the same thing and much more easily handled, the 
differences in the lengths of the addendum arcs. It is general in its nature, 
and independent both of the form of the tooth and of its length, which have 
but a minute effect on the required series. Any method that recognizes 
these small differences must necessarily require more intricate and 
difficult trigonometrical work than the slightly increased accuracy will war- 
rant.* 



* In his treatise on Kinematics, Prof. C. W. MacCord has treated my formula in such a summary 
and unjust manner, that in replying to him I do not feel bound by the usual rules of courtesy, but am at 
liberty to state the facts in plain words, without fear or favor. 

He not only refers to my process with an evident attempt at ridicule, but he positively mangles the 
facts. He is careful to show its defects and to exagerate their importance, while he is equally careful 
to slight and conceal its real merit. His motive is evident when he next proposes as a substitute a 
"locus" method which he claims is "the perfect solution of the problem," which will give a series 
that is "exact to a single tooth," and the value of which he assumes but does not attempt to prove. 
It is, in fact, an arbitrary approximation, and so wonderfully intricate, clumsy, and inaccurate, that the 
result, determined by it with great care by its own expert inventor, does not divide the locus curve to a 
single tooth, or in some parts, within several teeth, or distnbute the errors of form any more unifonnly 
than does the method it was intended to displace. A full (and free) discussion of this matter may be 
found in several letters published in the American Machinist in 1884. The locus method gives a result 
almost identical (for cases in actual use) with the series found by my formula, and if the slight differ- 
ence can be proved to be in its favor, as has not been done, it is of imperceptible importance, and no 
offset whatever to the excessive intricacy of the method. 

I did not claim perfection for my formula, or imagine it worth the notice that has been taken of 
it, and I would not in ordinary cases criticise the work of any other writer, but as I have been used 
with unusual and unprovoked seventy, I find it necessary to publish this note in self defence. Both 
sides of the question are now accessible to any one who may be interested, and all I ask or expect is 
that my work shall be treated with ordinary fairness, and allowed whatever merit it really has. 



28 



For the ordinary series of eight intervals, to cover fit>m 12 to oo the form- 
ula becomes 

^ 8-s 

and if we put s successively equal to 1, 2, 3, 4, 5, 6, 7, and 8, we get the 
series of last teeth 

13f , 16, 191, 24, 32, 48, 96 and oo , 
the resulting equidistant series being 

12 to 13, 25 to 32, 

14 to 16, 33 to 48, 

17 to 19, 49 to 96, 

20 to 24, 97 to a rack. 

Similarly, if we apply the formula from a=24 to z=x ,for n=12, we get the 
series adopted above for the involute odontograph table, and it requires but 
a few figures and a simple operation to apply it to any other case. 

POSITION OF THE "PERFECT" TOOTH. 

The "perfect" tooth, whose shape does duty for the whole interval, can 
best be placed, not at the center of the interval, but by assuming the inter- 
val to be a short series of two intervals, and adopting the intermediate 
value, The proper fi-xed shape for the interval from c to d is that of the 
tooth found by the formula 

^ c+d 

For the interval from 145 to 288 the perfect tooth is the 193rd, instead of the 
216th at the center. 



MAXIMUM ERROR OF THE SERIES. 

The odontograph gives the correct position of the perfect tooth only, and 
the point of the tooth at either end of the interval is out of position by the 
very small amount found by the formula 

• .182 

errors — 

pn 

in which p is the diametral pitch, and n is the number of intervals in the 
series. The odontograph table I have given for epicycloidal teeth has 
twelve intervals, and the greatest error in the position of the point of anj' 
tooth drawn by the table is 

error =— = '— inch. 

I2P p 

This becomes .015 inch for one diametral pitch, and .005 inch for one inch 
circular pitch and in direct proportion for other pitches. 
For involute teeth this formula becomes 

error = -2^ 

and for the given table having twenty-four intervals the greatest error is 
.006 inch for one diametral pitch, and .002 inch for one inch circular pitch. 

It is thus seen that the number of intervals used is sufficient for all prac- 
tical purposes, particularly if the error is still further reduced by adopting 
intermediate tabular numbers for intermediate numbers of teeth. 



29 



STANDARD FACES FOR GEAR WHEELS. 

It is desirable for the sake of uniformity and interchangeability, to have a 
regular system or law of fixed relation between the size of the teeth and the 
width of the face of a gear wheel. 

Such a law is recognized and in general use in a loose way, is a law of 
common sense, in fact, for it is almost invariably the custom to adopt a 
coarse tooth for a wide face, and although the practice is far from uniform, 
an examination of a great many cases, selected at random, will show that 
the "base," or product of the face and pitch, will average very near the 
number ten for cut iron gears. 

It is obvious that a fixed law should accommodate itself to actual practice 
as nearly as possible, and, adopting ten as a base as rigidly as a proper re- 
spect for standard pitches, and convenient fractions for the faces will permit, 
we can construct the following table for cut iron gears. 

Face Pitch Base 

i 20 10 

f 16 10 

f 12 9 

1 10 10 

li 8 10 

If 6 lOi 

2^ 4 10 

For small cut gears, which are usually made of brass, the weaker metal 
requires a coarser base, and we can use the number six for the standard. 

Face Pitch Base 

i 48 6 

i 24 6 

A 20 6i 

f 16 6 

A 14 6i 

i 12 6 

In the same way for cast gears we can construct a system on the number 
three as a base, as follows : — 

Face 

n 

2 
3 
4 

6 

e 

7 
8 
8 
9 



Circular Pitch 


Base 


i 


3 


i 


2f 


1 


3 


li 


H 


H 


3 


If 


2f 


2 


3 


2i 


H 


2i 


H 


2i 


2H 


8 


3 



WATER MOTORS 

F^or Driving a.11 descriptions of 
Snaall IVIachin^ery t>y common 
City Water F*ressu.re. 



Send for Pamphlet, and enclose stamp for reply. 
Full particulars should be given of the work to be 
done and the water pressure at command, 

I am preparing to put on the market a line of 

GEAR CnTTING MACHINERY, 

^vhieh I intend shall be equal to the best, and 
superior to most tools offered in this line. 

A large Universal Engine, to cut to six feet 
diameter, and entirely autoi"natic in its operations. 

A Medium Universal and Automatic Engine, to 
cut to three feet diameter. 

A Small Universal and Automatic Engine, to 
cut to one foot diameter. 

A cheap Gear-Cutting Attachment for common 
lathes. 

A Rack Cutter. 

A Rack-Cutting Attachment for ordinary gear 
cutters. 



Correspondence solicited with concerns that use and in- 
tend to purchase machinery of this description. 



THE 

CALCULATING MACHINE 



19 AN INSTRUMENT FOR THE 



-^CCTJI^.-A-I'E], E-^S-S", and. I^-^^>I3D 

Performance of the usually tedious operations of 

ADDITION, SUBTRACTION, MCTLTIPLIOATION, AND DIVISION. 




IT HAS BEOJBIVED THE 

M:. C. M. -A.. GS-old MCedal, 

M. C. M:. J^, Silver IMedal, 

rVixG Centennial M!edal, 
and the 
Soott Xjegaoy and IVIedal, 

AND IS ACKNOWLEDGED TO BE THE BEST INSTRUMENT IN USE FOR ITS PURPOSE. 



SEND STAMP FOR ILLUSTRATED PAMPHLET, 

giving full particulars of its construction and operation, and 

from prominent scientific and practical experts. 

PRICE, ONE HUNDRED DOLLARS. 

NO DISCOUNTS. NO A$£NTS WANTED. 



All descriptions of Gear Wheels made or cut to order. 

Any kind, Spur, Bevel, Miter, Rack, Ratchet, Worm, In- 
ternal, etc. 

Any size from a quarter inch to six feet diameter. 

Any quantity from a Single Gear to thousands. 

Gears for Machine Work, Gears for Model Work, Gears 
for Light or Heavy Machinery. 

BRASS AND IRON 

GEAR WHEELS 

GEAR CUTTING 

OF ALL DESCRIPTIONS. 



Send for IHuBtr&ted and Descripi/ve Pamphlet and Prhe Liet 



GEORGE B. GRANT, 

66 Beverly Street, BOSTON. 

Many sizes of ready-made Gears are kept in stock for 
immediate delivery. 

Brass Gears of all kinds for Models and light Machinery 
kept in stock and sent free by mail at low prices. 

Superior Spur, Bevel and Miter Gears with cast teeth. 

Cut Iron Gears, not ready-made, but that can be made to 
order at short notice from patterns and castings, always on 
hand. 



ODONTOGRAPHS. 



The new Odontographs, fully explained and discussed 
in the Handbook, are published in 

SEPARATE AND SIMPLE FORM, 

on heavy and durable paper, and, -with the tables of pitch 

diameters, clearances, etc., are arranged with special 

reference to 

PRACTICAL APPLICATION 

in the shop or drafting-room. 



IHor JN^srOUXJTJEl TKKXH, I'rloe, 25o. postpaid, 
mor EinCYCILiOIDAIj TEKTH, Prioe, S5o. postpaid. 
Both ODONTOG-RAFHS for 40o. 

I HAVE IN PREPARATION 

A SHOP IVIANXJAlv 

ON 

GEAR WHEELS. 

This will not interfere with the Handbook on the Teeth of 
Gears, or treat of Tooth Curves or Odontographs, but will deal 
"Brith the Gear Wheel as a whole and in connection with other 
Gears. 

It will fully describe and illustrate all the different Gears in 
common use, — the Spur, Miter, Bevel, Ratchet, Internal, Rack, 
and Worm Gear, showing what each one is, hovr it works, how it 
combines in pairs or trains with others, and how it is to be pro- 
portioned and shaped. Detailed examples with engravings to 
scale will illustrate each subject, and the object kept in view will 
not be to discuss or demonstrat>e, but to state facts in such plain 
tterms that they shall be readily understood by any intelligent 
mechanic. 

It will pay particular attention to the sizing and shaping of 
Miterand Bevel Gear Blanks, giving tables of angles and outside 
diameters, ready for shop use. Perfectly accurate dimensions and 
angles can be found at a glance, that usually require a careful 
4lrawing and close measurements. 

It will be uniform with the Handbook in size and price. 




mm,^3!^l OF CONGRESS 



021213^114 3 

GEORGE B. GRANT, 

66 BEVERLY. STREET, 

BOSTON, 

MASS. 



3 




GEAR CUTTING, 



Standard Gear Wheels, 

CALCULATING MACHINES, 
WATER MOTORS. 



